I've got the following expression and I want to know its truth value. What confuses me is the fact that False implies Truth is True. So, what happens if there's NO "a" value in the domain of some Interpretations? Shouldn't X=a be false? And therefore the expression inside the curved brackets would be true? Thinking it with intuition then P(a) cannot be true because there are some interpretations which make true the formula inside the brackets, where a is not in the domain.
{∀x∀x (x=a ⟹⟹ P(x))} ⊨⊨ P(a)
if we see it the other way around:
{∀x∀x (P(x) ⟹⟹ x=a)} ⊨⊨ P(a)
This one is clearly false, I[P] can be void and as False implies True is truth, the formula inside the brackets would be always true for those Interpretations (or models) who do not have any element inside de set of I[P] = {elements which satisfy the P property = 0}. In this case P(a) is clearly false.
So, can x=a be false in these formulas? I have to assume always that any constant is in the domain of all the interpretations which satisfy the formulae inside the curved brackets?
As Emil Jerabek said in the comments when you asked this question at MO, yes - every constant symbol names an element of the domain. Similarly, every function symbol names a function from (a Cartesian power of) the domain to the domain, and so forth.
Note that this is explicitly covered in the definition of "interpretation" (although some texts like http://library.msri.org/books/Book39/files/marker.pdf gloss over it - there, it's implicit in the phrase "An L-structure is a structure M where we can interpret all of the symbols of L" (p. 16)): for example, see Definition 1.2 of https://www3.nd.edu/~cmnd/programs/cmnd2016/undergrad/Conant_MTnotes.pdf.